 Solution

Decompose $\mathbf{v}$ by orthogonal projection onto the straight line spanned by $\mathbf{n}$ and its orthogonal complement:The first term is fixed by the rotation. If $\mathbf{w}\ne 0$, the vectors $\mathbf{w}/|\mathbf{w}|$ and $(\mathbf{n}\times \mathbf{w})/|\mathbf{w}|$ form an orthonormal basis of the perpendicular plane, oriented in the positive sense about $\mathbf{n}$. A planar rotation through $\theta$ therefore sendsHere $\mathbf{n}\times \mathbf{w}=\mathbf{n}\times \mathbf{v}$. Adding the fixed parallel component proves the Rodrigues rotation formula in the required form:When $\mathbf{w}=0$, both perpendicular terms vanish, so the same formula fixes $\mathbf{v}$ as required. Positive angles use the right-hand convention about $\mathbf{n}$.